Abstract
d’Aspremont (Econometrica 47:1145–1150 , 1979) showed that a Hotelling (Econ J 39:41–57 , 1929) duopoly model with quadratic transport costs yields maximal differentiation. However, the introducing of an online firm ensures that the duopolist will never be located at the end points of the market. In other words, an online firm can raise a market effect that induces two firms to be finitely differentiated. The implication of the socially optimal solution is derived. The results herein can be extended to allow multiple firms. Finally, a free-entry equilibrium and the Stackelberg equilibrium are also discussed.
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Notes
As an extensive application of Hotelling-like models, Wolinsky (1987) also provides a related framework in which labeled products and unlabeled products in a traditional industrial organization can be regarded as goods sold by retail stores and the online store, respectively.
Besides the waiting cost, consumers cannot touch, hear or smell the products that are listed on a Web site. Hence, the consumers may be uncertain regarding the quality of the product. This fact is an obvious disadvantage for the online purchases.
Solving the first-order condition yields another solution: \(p_1 = p_2 =-\frac{2z}{5}+\frac{k}{100}+\frac{\sqrt{k(k+80z)}}{100}, p_3 =-\frac{2z}{5}+\frac{3k}{200}-\frac{3\sqrt{k(k+80z)}}{200}<0\), which results in a negative equilibrium price set by the online firm, and so this solution is excluded.
In this situation, only firm 1 moves from the interior locations to a boundary location. Alternatively, another symmetric situation such that \((x_1 =x_1^*,x_2 =1)\) is similar and is thus omitted here.
However, if only one brick-and-mortar firm is competing with one online firm, a boundary location may be better than the interior location for the brick-and-mortar firm when \(z\) is small enough. Specifically, when \(\frac{z}{k}<\frac{1}{5}[(2^{1/3}+1)^{2}-1]\cong 0.8214\), the boundary location is better for firm 1 than is the interior location. Similarly, when the transportation cost is linear in distance, the boundary location is better for firm 1 than is the interior location when \(z^{2}<\frac{k^{2}}{2}\) because \(\pi _1 (x_1 =0)-\pi _1^*(x_1 =1/2)=\frac{(k^{2}-2z^{2})}{18k}\). In fact, Anderson (1988) shows that when the transport cost function is linear quadratic, there is no pure strategy equilibrium whenever the parameter of the linear part is not zero. We are grateful to one of the anonymous referees for pointing out the case of corner locations.
It is possible to impose price regulation to prevent spatial duopolistic firms from engaging in maximal product differentiation. For instance, Bhaskar (1997) introduces a price floor into the Hotelling (1929) model with quadratic transport cost functions and shows that the minimal differentiation equilibrium exists if and only if the price floor is high enough.
In our setting, the online firm cannot force out all brick-and-mortar firms, since they have the advantage of lower production costs for selling to nearby consumers.
Empirically, Clay et al. (2002) show that online bookstores and physical bookstores do not generally charge the same prices.
We are grateful to one of the anonymous referees for pointing out this game structure.
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Acknowledgments
The authors would like to thank the editor-in-chief, Börje Johansson and two anonymous referees for their valuable comments.